Theorem tpssd 32686

Description: Deduction version of tpssi : An unordered triple of elements of a class is a subset of that class. (Contributed by Thierry Arnoux, 2-Nov-2025.)

Hypotheses

Ref	Expression
tpssd.1	⊢ (𝜑 → 𝐴 ∈ 𝐷)
tpssd.2	⊢ (𝜑 → 𝐵 ∈ 𝐷)
tpssd.3	⊢ (𝜑 → 𝐶 ∈ 𝐷)

Assertion

Ref	Expression
tpssd	⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ⊆ 𝐷)

Proof of Theorem tpssd

Step	Hyp	Ref	Expression
1		tpssd.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐷)
2		tpssd.2	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝐷)
3		tpssd.3	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝐷)
4		tpssi 4795	. 2 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → {𝐴, 𝐵, 𝐶} ⊆ 𝐷)
5	1, 2, 3, 4	syl3anc 1389	1 ⊢ (𝜑 → {𝐴, 𝐵, 𝐶} ⊆ 𝐷)

Syntax hints:   → wi 4   ∈ wcel 2141   ⊆ wss 3904  {ctp 4585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-un 3909  df-ss 3921  df-sn 4582  df-pr 4584  df-tp 4586

This theorem is referenced by:  constrlccllem  34011  constrcccllem  34012  cos9thpiminplylem2  34041